Several nonlinear dynamical systems exhibit non-isochronous oscillations. In these oscillators one can see that the frequency of oscillations depends on the amplitude. For example, the anharmonic oscillator equation \(\ddot{x}+c_1x+c_2x^3=0\) , with \(c_1>0\) , \(c_2>0\) , admits the solution \(x(t)=Acn(\omega t,k)\) , \(\omega =\sqrt{c_1+c_2A^2}\) , where \(cn\)  is the Jacobi elliptic function of modulus \(k=\sqrt{c_2A^2/2(c_1+c_2A^2)}\) . While deriving this solution we have considered the initial conditions \(x(0)=A\) , \(\dot{x}(0)=0\)  [1]}. Unlike the linear harmonic oscillator equation the angular frequency \(\omega \)  is now not a constant but depends on the amplitude \(A\) . These non-isochronous dynamical systems exhibit a variety of dynamics differing from the isochronous systems. However, a class of non-isochronous dynamical systems also admit periodic oscillations of harmonic type. A notable example in this category is the Mathews-Lakshmanan nonlinear oscillator equation, \((1+\lambda x^2)\ddot{x}-\lambda x\dot{x}^2+\omega _0^2x=0\) , where \(\lambda \)  and \(\omega _0\)  are constants [2]}, [3]}. This nonlinear/nonpolynomial oscillator equation has the solution \(x(t)=A\cos \Omega t\) , \(\Omega =\omega _0/\sqrt{1+\lambda A^2}\) . We have obtained this solution for the initial conditions \(x(0)=A\) , \(\dot{x}(0)=0\)  [1]}. The angular frequency (\(\Omega \) ) is amplitude dependent and hence the Mathews-Lakshmanan oscillator is a non-isochronous system. Because of the amplitude dependent frequency property, various responses have also been exhibited for different input conditions by the underlying dynamical system. This kind of nonisochornicity nature can be seen in various physical and biological systems. In particular, one finds that strong nonisochronicity produces various synchronization patterns as well as long irregular transient phase dynamics in networks of such oscillators in the case where the long transient dynamics has connection with the functioning of biological systems [5]}, [6]}, [7]}.
